For multi-qubit entangling operations, a popular and widely used two-qubit entangling gate for optical qubits is known as the Mølmer–Sørensen (MS) gate [1]}, [2]}, [3]}. This gate equips the static string with complete qubit connectivity, as it exploits a common vibrational mode as a quantum data bus, which can mediate the entangling gate operation between an arbitrary pair (or larger groups) of qubits that share this common mode, provided that the laser beams uniformly illuminate this pair (or group) and do not illuminate the remaining ions.
For this work, we concern ourselves only with the MS gate in the \(XX\) -basis which, when addressed to the pair of ions with indexes \(i,j\)  within the static string, can be expressed through the following unitary
\(XX_{ij}(\theta ) = \exp \left( -i \frac{\theta }{2} X_iX_j \right),\) 
